The example I gave on my blog and in the article was a cylinder that looked like a circular cylinder but like a square cylinder in the mirror. You can download a printable pdf template to make your own, and you can watch this video that I made.

Today I designed two more impossible cylinders. The first looks like a square cylinder and like a triangular cylinder in the mirror. Download a printable pdf template to make your own.

I also designed one that looks like a heart-shaped cylinder and like a diamond-shaped cylinder in the mirror. (The spade/club one is an exercise for the reader.) Here’s a template to make your own.

Update: I just created another cylinder. I’m not as pleased with this one. The shape of the cylinder is complicated enough that it is difficult to get it to look right. Here’s my best attempt at a star/moon cylinder. (Pdf template.)

In this blog post we show how he achieved the effect. For simplicity, we will show how he made a shape that looks like a circular cylinder from the front and a square cylinder in the mirror.

The following applet shows our final product (clicking the image links to the GeoGebra applet). It is a closed curve that represents the top rim of Sugihara’s shape. You can rotate the axes with your mouse. If you view the coordinate system with the positive green and blue axes lined up (1 with 1, 2 with 2, and so on), the curve will look like the unit circle in the green-red plane. If you drag the image so that the positive blue axis lines up with the negative green axis (1 with -1, 2 with -2, and so on), it will look like you are viewing a square (oriented as a diamond) in the green-red plane.

Here are screenshots showing the two views.

How does it work? It is all about perspective.

To set this up mathematically, we imagine two viewers in 3-dimensional space. One viewer is at and the other is at (in the video this second viewer is you, in the mirror). They are looking down on a curve However, from their vantage points it looks like they are seeing two different curves in the -plane: and respectively.

In our example the two observed curves are the unit circle and the square passing through the points , as shown in the -plane below. We will have to break each of these shapes into two different curves, so we’ll have and Also, we could choose to be some suitably large number greater than 1, but in fact, as we will see, taking the limit as tends to infinity produces a lovely final expression. For now we will continue to work in generalities and will wait to insert these specifics later.

Our aim is now to define Let’s fix , and let and be two points on the curves in the -plane. In order for the person at to view her shape, must lie on the line (see figure below). Likewise, for the person at to see his shape, must lie on the line Thus, must be the point of intersection of lines and (We know that the lines are not skew because they lie in the plane containing the points and and for appropriate choices of and the lines intersect and the point of intersection is below

It is straightforward to show that is a parametrization of the line and is a parametrization of A little algebra shows shows that their point of intersection is

and thus our desired curve is

Because is a large value, we can take the limit as goes to infinity. This yields the elegant expression

We may now plug in our functions. The portion of our curve with nonnegative -coordinates is given by

for and the other half by

for This is the curve shown in the applet.

[Update: See the comment by Joshua and my reply for a simpler way of obtaining the parametrization.]

I’d like to thank Matt Parker for introducing me to diameter tapes (or D-tapes). These are measuring tapes used by foresters to measure the diameters of trees. The forester wraps the measuring tape around a tree as if measuring the circumference, but the scale on the tape is adjusted so that the measurement gives the diameter (assuming the tree has a circular cross section). It uses the fact that C=πD.

Parker mentioned the existence of these measuring tapes in his talk at Gathering 4 Gardner 12 and he gave every attendee a D-tape that he made (see below).

That got me thinking. If you know the circumference, you can compute the diameter. But you can also compute the area. Also, if you wrap the tape around the equator of a sphere, you can compute the the volume and the surface area of the sphere.

Thus, inspired by this, I made measuring tapes to measure the diameter and area of a circle and the volume and surface area of a sphere. The top of each tape is marked off in centimeters, so it measures the circumference. The bottoms can be used to measure the diameter of the circle, the cross-sectional area, the volume of the sphere, and the circumference of the sphere.

I’ve also created a printable pdf of these measuring tapes, which should, I hope, print to the right scale so that 1 cm on the measuring tape is actually 1 cm. (Note that although a 25 cm ruler looks long, when you wrap it around a circle, you find that the diameter is only 8 cm. Not very big.)

It was a fun exercise to figure out how to mark and make the rulers. Except for the diameter ruler, the relationships aren’t linear. (I set everything up in Excel, then I moved the info over to a Geogebra spreadsheet. I used GeoGebra to draw everything. I exported the images as pdfs. Then I cleaned them up and added the text in Adobe Illustrator.)

I’ve taught topology many times. One of the highlights for the students (and for me) is the investigation of the Möbius band—the one sided, one edged, non-orientable surface with boundary. On the day we introduce the Möbius band I bring many strips of paper, clear tape, and scissors and have the students make conjectures about what would happen if we taped and cut apart various topological shapes. Here are some activities that are fun to do:

Twist the paper zero times, and tape the ends (making a cylinder). Cut down the midline.

Give the paper one half-twist, and tape the ends (making a Möbius band). Cut down the midline.

Give the paper two half-twists ,and tape the ends. Cut down the midline.

Give the paper three half-twists, and tape the ends. Cut down the midline.

When I got home I bought zippers and Velcro and used them to make reusable strips. Half of them were just one zipper with Velcro on the ends. These can be used to do activities 1-4 above. Then I took identical pairs of zippers and joined them side-by side to make strips for activities 5-8.

Some tips:

The zippers I used were 12″ or 14″ long. They seemed to work well.

The Velcro has adhesive on the back, so I could just stick them to the ends of the strips. I did it so that the velcro folded over the ends and so was on both sides. That gave me flexibility if I gave an even or an odd number of half twists.

For the doubled zippers, align them so that both zippers start at the same end of the strip and so that one is on the top side and one is on the bottom side. Then, when you join with an odd number of half-twists, one begins where the other ends and they’re both on the same “side” (locally) of the Möbius band.

Ideally I would have sewn the two zippers side-by-side. But we don’t own a sewing machine. So I stapled them.

I learned the hard way that there are separating zippers (think of the zipper on a parka that comes apart at the bottom) and closed-end zippers (think the zipper on your pants that stops at the bottom). You want the former type for this activity.

The zippers were about $3 apiece, but my wife told me about a local fabric store that was going out of business, so I got them for 70% off!

I spent a good chunk of last week reading about David Johnson Leisk (1906–1975), who is better known by his nom-de-plum Crockett Johnson. Johnson is most well known as the author of Harold and the Purple Crayon, a children’s book from 1955, and its sequels. Johnson was also the author of the 1940s comic Barnaby.

Later in his life Johnson became interested in mathematics. He was particularly interested in geometry, and most specifically in the problems of antiquity (squaring the circle, trisecting the angle, doubling the cube, and constructing regular polygons). He turned many geometric theorems into works of art. Eighty of his paintings are now at the Smithsonian.

Johnson even created new mathematics. I would like to discuss one of his contributions here. (See his article “A Construction for a Regular Heptagon,” The Mathematical Gazette Vol. 59, No. 407 (Mar., 1975), pp. 17-21.)

The heptagon is notewothy because it is the regular polygon with the fewest number of sides that cannot be constructed with compass and straightedge alone. In his article, Johnson gives a way to construct the heptagon using a marked straightedge (this is called a neusis construction). Johnson did not give the first neusis construction of a heptagon—François Viète gave the first such construction in 1593. (Also, Archimedes gave an unorthodox neusis-like construction).

However, Johnson’s proof used trigonometry (including the law of cosines and several trigonometric identities). My question to you is: Is there a purely geometric proof of his result? I played around with it for a little while and couldn’t find one, and I couldn’t find a geometric proof in the literature.

The construction

The key to Johnson’s construction is producing 3:3:1 triangle; that is, a triangle in which the angles are in a 3:3:1 ratio (they would be and The three vertices of the triangle are three vertices of a regular heptagon. If we construct the circumcircle, then it is easy to construct the four remaining vertices with a compass and straightedge.

Here’s his construction of a 3:3:1 triangle using a marked straightedge—that is, an otherwise ordinary straightedge, but possessing a mark one unit from the end (or equivalently, two marks one unit apart). We begin by drawing a line segment AB of length one (see below, left). Construct a unit line segment AC perpendicular to AB. Also, construct the perpendicular bisector to AB; call it l. Then construct a circle with center B and radius BC. Now we perform the neusis construction with the marked straightedge: Construct a line AD so that D is on l and D is one unit from the circle. (That is, the end of the straightedge is at D, the mark is on the circle, and the edge passes through A.) Then is the 3:3:1 triangle.

A geometric proof?

Johnson’s proof that is a 3:3:1 triangle used trigonometry (see his article for details). Is there a geometric proof?

Boiled down to its essence, here’s the question: Suppose is isosceles and E is on AD. Moreover, suppose and , prove that

Here is a fact that may help. Johnson discovered this fact—ironically—when he was sitting in a café in Syracuse on Sicily (the birth place of Archimedes) playing with the wine and food menus and some tooth picks. If we have an isosceles triangle ABD with the point E on BD and F on AD such that then triangle ABD is a 3:3:1 triangle. (This is easy to prove. Suppose Then, because is isosceles, . So the exterior angle of is Because is isosceles, Lastly, observe that and are similar, so It follows that )

Thus, a possible route to proving the theorem is to find a point F in the original diagram so that

If you find a proof, I’d love to hear about it!

Here are two of the paintings Johnson made from his work with the heptagon.

Abstract: In 1928 Henry Scudder described how to use a carpenter’s square to trisect an angle. We use the ideas behind Scudder’s technique to define a trisectrix—a curve that can be used to trisect an angle. We also describe a compass that could be used to draw the curve.

I just read this post at Futility Closet. (Spoiler: Don’t click the link unless you want to know the punchline.) Perhaps the result is well known, but I hadn’t seen it before.

The post made me think of a neat project for an “Introduction to proofs” class. I’ll have to save it for the next time I teach that class (Discrete Mathematics at my college).

Start with the first n prime numbers, Divide them into two sets. Let A be the product of the primes in one set and let B be the product of the primes in the other set (the product is 1 if the set is empty). For example, if n=4 we could have {2,3,7} and {5}, so A=42 and B=5. What can we say about A+B and A-B? Form a conjecture and prove it.

I like the problem because I could imagine the students will go through a process of (1) I have no idea. (2) Aha! I’ve figured it out. How do I prove it? (3) Wait, I found a counterexample. (4) Ahhh! I see what’s going on, here’s the theorem, and here’s the proof. (I think they would come up with a different statement of the theorem than the one in the Futility Closet post.)

I’m teaching Discrete Math this semester. Discrete Math is our college’s “introduction to proofs” class. We spend a lot of time talking about and practicing proofwriting. In earlier blog posts I shared my “Nuts and Bolts of Writing Mathematics” and an “editing checklist” that I give to them.

Yesterday, I gave them an example of a very, very poorly written proof that I created. (It was fun trying to insert as many problems as I could into the short proof!) I had them work on the assignment in class for about 7 minutes. Then we came together and talked about all the errors that we could find, and finally we rewrote the proof in the correct way. Here’s a pdf of the assignment, and I’ve posted a screenshot below.

In case you are wondering, here are the errors in the proof. Numbers refer to sentence numbers.

The proof should be begin immediately after the word PROOF.

The entire proof should be written in paragraph form.

1. Capitalize the first word of the sentence.

1. Split this sentence in two: The first sentence should be “Let m be an odd integer and n be an even integer.” In the second sentence we apply the definition of even and odd.

1. Do not use k for both the even and the odd numbers.

1. Don’t use a symbol (#) in place of a word.

2. Start the sentence with a word, not a mathematical expression.

2. The correct expression is mn2, not m2n.

2. “Equivalent to” is a term we use for logical expressions, not numbers. It should be “equal to”—and in fact, we should just use “=.”

2. Algebra error: the term 8k should be 8k2.

2. In a 200-level math class we do not have to show all of these algebraic details.

3. This is not how we show that a number is even—we must show that mn2 satisfies the definition of even.

3. Do not use the word “obviously.”

3. In a proof like this we’d have to show that the three terms satisfy the definition of even.

3. “There” should be “their.”

3. Missing dollar signs around 2k in the LaTeX code.

4. Do not use the passive voice.

4. Missing period at the end of the sentence.

4. and 5. Write in present tense.

5. Do not put an example in a proof.

5. There are missing dollar signs in the LaTeX : after m=5 and before n=4.

5. It should be first person plural, not first person singular (“we” not “I”).